class: center, middle, inverse, title-slide # Cencored regression, weak IV, and quantile regression ## Tutorial 1 ### Stanislav Avdeev --- # Tutorials * 7 TA sessions * 6 TA sessions are about lecture material * The last session is primarily about exam and remaining questions about the course material (TBA) * Send me any questions you want to discuss before each TA session * Use Canvas or send me an email * Alternately, leave your questions anonymously here: https://www.menti.com/c6uyd9qan4 (I will update the link each week on Canvas) --- # Assignments * Due date: 11:59pm on Sundays (the first assignment is an exception: 11:59am on Tuesday) * Assignments are graded within a week from the deadline * Solutions will not be shared so if you want to discuss a specific exercise, let me know before the TA session (you submit your solutions on Sunday, thus, we can discuss any questions on the following TA session on Tuesday) --- # Course objective * The key objective of the course is **applying** microeconometric techniques rather than **deriving** econometric and statistical properties of estimators. * In other words, there’s way less of this: `\begin{align*} \text{plim} \hat{\beta}_{OLS} = \beta + \text{plim} (\frac{1}{N}X'X)^{-1} \text{plim} \frac{1}{N} X' \epsilon = \beta + Q^{-1} \times \text{plim} \frac{1}{N} X' \epsilon \end{align*}` * And way more of this: ```r library(fixest) tb <- tibble(groups = sort(rep(1:10, 600)), time = rep(sort(rep(1:6, 100)), 10)) %>% mutate(Treated = I(groups > 5) * I(time > 3)) %>% mutate(Y = groups + time + Treated*5 + rnorm(6000)) m <- feols(Y ~ Treated | groups + time, data = tb) ``` If you would like to go deeper into the former, take Advanced Econometrics I and II next year --- # Weak instrument problem - Weak instrument problem means that we probably shouldn't be using IV in small samples - This also means that it's really important that `\(cov(X, Z)\)` is not small - If `\(Z\)` has only a trivial effect on `\(X\)`, then it's not *relevant* - even if it's truly exogenous, it does not matter because there's no variation in `\(X\)` we can isolate with it - And our small-sample bias will be big --- # Weak instrument problem - There are some rules of thumb for how strong an instrument must be to be counted as "not weak" - A t-statistic above 3, or an F statistic from a joint test of the instruments that is 10 or above - These rules of thumb aren't great - selecting a model on the basis of significance naturally biases your results - What you really want is to know the *population* effect of `\(Z\)` on `\(X\)` - you want the F-statistic from *that* to be 10+. Of course we don't actually know that. --- # Weak instrument problem: simulation - Let's look at the output of `feols()` using a simulated dataset ```r library(fabricatr) set.seed(777) df <- fabricate( N = 200, Y = rpois(N, lambda = 4), Z = rbinom(N, 1, prob = 0.4), X1 = Z * rbinom(N, 1, prob = 0.8), X2 = rnorm(N), G = sample(letters[1:4], N, replace = TRUE) ) ``` --- # Weak instrument problem: simulation ```r iv <- feols(Y ~ X2 | X1 ~ Z, data = df, se = 'hetero') thef <- fitstat(iv, 'ivf', verbose = FALSE)$`ivf1::X1`$stat iv ``` ``` ## TSLS estimation, Dep. Var.: Y, Endo.: X1, Instr.: Z ## Second stage: Dep. Var.: Y ## Observations: 200 ## Standard-errors: Heteroskedasticity-robust ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 3.616743 0.164506 21.98548 < 2.2e-16 *** ## fit_X1 1.066064 0.369839 2.88251 0.0043831 ** ## X2 0.326317 0.153746 2.12244 0.0350497 * ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## RMSE: 1.93622 Adj. R2: 0.009568 ## F-test (1st stage), X1: stat = 522.5, p < 2.2e-16 , on 1 and 197 DoF. ## Wu-Hausman: stat = 11.1, p = 0.001029, on 1 and 196 DoF. ``` - 522.47 is way above 10! We're probably fine in this particular regression --- # Overidentification tests - "Overidentification" just means we have more identifying conditions (validity assumptions) than we actually need. We only need one instrument, but we have two (or more) - So we can compare what we get using each instrument individually - If we assume that *at least one of them is valid*, and they both produce similar results, then that's evidence that *both* are valid --- # Overidentification tests: simulation - We can do this using `diagnostics = TRUE` in `iv_robust` again ```r set.seed(1000) # Create data where Z1 is valid and Z2 is invalid df <- tibble(Z1 = rnorm(1000), Z2 = rnorm(1000)) %>% mutate(X = Z1 + Z2 + rnorm(1000)) %>% # True effect is 1 mutate(Y = X + Z2 + rnorm(1000)) iv <- feols(Y ~ 1 | X ~ Z1 + Z2, data = df, se = 'hetero') fitstat(iv, 'sargan') ``` ``` ## Sargan: stat = 267.8, p < 2.2e-16, on 1 DoF. ``` - That's a small p-value! We can reject that the results are similar for each IV, telling us that one is endogenous (although without seeing the actual data generating process we couldn't guess if it were `\(Z1\)` or `\(Z2\)` ) --- # Overidentification tests: simulation - How different are they? What did the test see that it was comparing? ```r iv1 <- feols(Y ~ 1 | X ~ Z1, data = df) iv2 <- feols(Y ~ 1 | X ~ Z2, data = df) export_summs(iv1, iv2, statistics = c(N = 'nobs')) ``` <table class="huxtable" style="border-collapse: collapse; border: 0px; margin-bottom: 2em; margin-top: 2em; ; margin-left: auto; margin-right: auto; " id="tab:unnamed-chunk-5"> <col><col><col><tr> <th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Model 1</th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Model 2</th></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(Intercept)</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.01    </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.00    </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.04)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.05)   </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">fit_X</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">1.08 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">1.92 ***</td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.04)   </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.05)   </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">N</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.8pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">1000       </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.8pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">1000       </td></tr> <tr> <th colspan="3" style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> *** p < 0.001; ** p < 0.01; * p < 0.05.</th></tr> </table> Notice the first model gives an accurate coefficient of 1